CHAPTER 3 MORPHOLOGICAL PROCESSING OF DIGITAL IMAGES

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1 20 CHAPTER 3 MORPHOLOGICAL PROCESSING OF DIGITAL IMAGES INTRODUCTION Here the mathematical morphological operations of dilation, erosion, opening and closing are explained in brief along with their application to 2-D and 3-D binary and grey images. Morphology is a branch in biology that deals with the form and structure of animals and plants. Mathematical Morphology is a tool for extracting image components and it is useful in the representation and description of region shape. The language of mathematical morphology is Set theory. 3.1 PRELIMINARIES OF MATHEMATICAL MORPHOLOGY The basic operations of mathematical morphology are the dual operations of Dilation and Erosion. These two operations and their combinations allow one to modify the form and shape of a digital image. An image called Structuring Element decides the shape of the given image. The given image and the structuring element are viewed as sets. The union and intersection operation of these two sets are termed as Dilation and Erosion referred from Gonzalez and Woods, Prentice Hall et al (2002)., Gonzalez, R.C. Reading et al (1992)., Junichiro Toriwaki, Springer, et al (2009)., Pratt, W.K., New York, et al (1978).

2 21 Dilation is the operation that combines two sets using addition of set elements. Let A and B are subsets in 2-D space. Let A be the image to be processed and B the Structuring element. Now, the operation of dilation is expressed by Serra, J., Academic Press, London as 2 A B {c Z c a b for some a A, b B} (3.1) Where as erosion is the operation that combines two sets using subtraction of set elements and expressed as 2 A B {c Z c a b for some a A, b B} (3.2) The size of the image A is M N where M represents the width of the image and N represents the height of the image. The size of the image B (structuring element) is m n where m represents the width of the image and n represents the height of the image. For all practical purposes the height and width are same and is an odd number because point of reference is mid point. For example the structuring B could be of size 3 3, 5 5, 7 7 etc. The symbol a represents an element (coordinates) of image A and b represents an element of image B. For example, the equation c=a+b means that a=(x 1,y 1 ), b=(x 2,y 2 ), c=(x 3,y 3 ) and that x 3 = x 1 +x 2, y 3 = y 1 +y Minkowski Algebra Minkowski addition Consider two images A and B (sets) in Z 2, we define the minkowski Sum A B by as

3 22 A B A b b B (3.3) Minkowski addition is obtained by translating A by element of B and then taking the union of all the resulting translates. We confirm from minkowski addition 1. A + {(0,0) } = A 2. A + {x} = A + x for any point x in Z 2. Minkowski Subtraction subtraction as Consider two images A and B in Z 2, we define minkowski A B A b b B (3.4) Minkowski subtraction is obtained by translating A by every element of B and then intersection is performed. Equations 3.3 and 3.4 are referred from Serra, J., Academic Press, London ALGEBRAIC PROPERTIES OF DILATION AND EROSION 1. Commutativity The operation of dilation is commutative that is A B B A (3.5) The Commutative property states that forming the union of translates of A by elements of B is equivalent to forming the union of

4 23 translates of B by elements of A. Where A is treated as image and B as structuring element. But in general erosion is not commutative that is A B B A (3.6) 2. Associatively The operation of dilation is associative where as erosion is not. Minkowski sum (dilation) of any finite number of input images can be allowed in associative law without worrying about which is performed first. A (B C) (A B) C (3.7) 3. Translation invariance This property states that, translation followed by dilation is equivalent to dilation followed by translation. x is any point in Z 2. A (B x) (A B) x (3.8) 4. Distributivity i. Dilation distributes over union from both the right and the left as below. A ( B C) ( A B) (A C) (3.9) (( B C) A) ( B A) (C A) (3.10)

5 24 Dilating by the union is equivalent to dilating repeatedly and then taking the union of the resulting outputs. A, C are images and B is ii. Dilation does not satisfy the distributive property over intersection, but we can have A B C ( A B) (A C) (3.11) (( B C) A) ( B A) (C A) (3.12) iii. Minkowski subtraction satisfies a sort of left antidistributivity over union. We have A (B C) (A B) (A C) (3.13) In terms of erosion we write above equation as E A,B C A (-B) (-C) [A ( B)] [A (-B)] (A, B) (A,C) (3.14) Thus eroding by the union is equivalent to eroding by each structuring element independently and then intersecting the two outputs. iv. Also eroding the intersection of two sets by a given structuring element produces the same output as first eroding each set separately and then intersecting the results. (B C) A (B A) (C A) (3.15)

6 25 5. Duality Dilating an image can be obtained by eroding the complementary image, and eroding an image can be obtained by dilating the complementary image. C C [D (A, B) ] (A B) C C [ (A, B) ] D (A B) (3.16) 6. Extensivity If 0 B, A A B implies Dilation and erosion are increasing operations that is if A B A C B C (3.17) A C B C (3.18) An operation on images in the plane say P(A) is said to be increasing if, whenever A is a sub image of B, then P(A) is a sub image of P(B). P is said to be decreasing if A B means P(A) P(B). In morphological analysis increasing sets play an important role especially in morphological filters. i. If B is fixed and A 1 A 2 then D ( 2 A1, B) D (A, B) (3.19) ii. If B is fixed and A 1 A 2 then

7 26 ( 2 A1, B) (A, B) (3.20) The above properties state that for fixed image B, dilating or eroding a larger image by a fixed image results in a larger output image. iii. If A is fixed and B 1 B 2 then ( A, B 1) (A, B2) (3.21) This property states that erosion is decreasing in the second variable. Thus it is clear that eroding a fixed image by a smaller image results in a greater output than eroding the same image by larger image, because the smaller structuring element fits comfortably Algebraic Properties Of Opening and the closing. Dilation and Erosion are the fundamental morphological operations, In addition to these, they are two other operations called opening and the closing. These are secondary operations. In terms of erosion and dilation, we have O ( A, B) D (E (A, B)) C ( A, B) (D (A, - B)) (3.22) The opening satisfies O(a,b) is a sub image of a (antiextensivity) If a 1 is a sub image of a 2, then O(a 1,b) is a sub image of O(a 2,B) (increasing monotonicity) O[O(a,b),b] = O(a,b) (idempotent)

8 27 Antiextensivity states that opening an image produces an output that is a sub image of original image. Increasing monotonicity states that, given a fixed structuring element, the opening is an increasing image to image mapping in the first variable. Idempotent states that successive openings by the same structuring element do not alter the image after the primary application. The above algebraic properties of the opening are used for the construction of morphological filters. Opening acts as a filter, the exact result depends upon the shape of the structuring element. The closing satisfies a is a sub image of C (a,b) (extensivity) If a 1 is a sub image of a 2, then C(a 1,b) is a sub image of C(a 2,B) (increasing monotonicity) C[C(a,b),b] = C(a,b) idempotent 3.3 MORPHOLOGICAL FILTERS OF OPENING AND CLOSING As outlined earlier, the basic operations of mathematical morphology are the two dual operations of Dilation and Erosion. These two operations and their combinations allow one to modify the form and shape of a digital image. If we denote the operation of dilation by the symbol 1 and that of erosion by 0, then 10 represents the morphological filter of closing. Similarly 01 represents the filter of opening.

9 28 The operations of closing and opening are called Morphological Filters. Closing and opening are idempotent operators, that is, closing or opening of an already closed or opened image with the same structuring element will not produce any change in the image. Each morphological filter is a series of dilation/erosion operations and is denoted by a string of binary digits. Binary digits 0 and 1 represent erosion and dilation respectively. The binary strings 10 and 01 represent closing and opening respectively as suggested by Rajan, E.G. U.K., Filtering of an image with 1010 produces the same effect as that with 10. Hence, the filter 1010 is termed as an Invalid Filter. A filter which is not invalid is called a Valid Filter. The filter is valid but the filter is invalid because is the same as Likewise one can construct valid filters denoted by binary strings of any length. Identification of invalid filter and replacing them with valid filters improves the overall performance of the image processing system. One can generate 2n C n valid filters from a string of 0s and 1s of length n. 3.4 MORPHOLOGICAL PROCESSING OF 2-D BINARY IMAGES Binary images are images whose pixels have only two possible intensity values. In binary images, the set elements are members of the 2-D integer space Z 2 where each element (a,b) is a coordinate of a black (or white) pixel in the image. Advantages of using binary images Smaller memory requirements Faster execution time

10 29 Many techniques developed for these systems are also applicable to vision systems which use gray scale images Less expensive Disadvantages of using binary images Limited application Losing internal details of objects (i.e. in inspection tasks) difficult to control the contrast between the background and the objects (i.e. in material handling and assembly tasks) Some aspects of Binary Vision Systems Formation of binary images: Binary image is obtained by thresholding the gray scale image. Many cameras are designed to perform thresholding in hardware. Geometric Properties: Size, Position, Orientation and Projection. Topological Properties Object recognition in binary images Morphological Erosion Of 2-D Binary Images used for Erosion shrinks the connected sets of 1 s of a binary image. It can be 1. Shrinking shapes

11 30 2. Removing bridges, branches and small protrusions Erosion is based on Minkowski Subtraction. A and B are two sets, then the Minkowski Subtraction is given by A (-) B A b b B (3.23) and is not commutative. That is, A (-) B B (-) A (3.24) Example : If A = {(1,0), (1,1), (1,2), (0,3), (1,3), (2,3), (3,3), (1,4)} and B = {(0,0), (1,0)} then the erosion operation is given by A(-)B = {(0,3), (1,3), (2,3)}as shown in the Figure 3.1 Figure 3.1 Example of Binary Erosion

12 Morphological Dilation Of 2-D Binary Images be used for Dilation expands the connected sets of 1 s of a binary image. It can 1. Expanding shapes 2. Filling holes, gaps and gulfs Dilation is based on Minkowski Addition. A and B are two sets, then the Minkowski Addition is given by A ( ) B (A b) b B (3.25) It is commutative. That is, A ( ) B B ( ) A (3.26) Example for dilation: If A = {(1,0), (1,1), (1,2), (2,2), (0,3), (0,4)} and B = {(0,0), (1,0)} then the dilation operation is given by A (+) B = {(1,0), (1,1), (1,2), (2,2), (0,3), (0,4), (2,0), (2,1), (3,2), (1,3), (1,4)}as shown in Figure 3.2

13 32 Figure 3.2 Example of Binary Dilation In a binary image, the foreground pixels are represented by 1's and background pixels by 0's as shown in Figure 3.3 Figure 3.3 Representation of Binary Image Algorithm For Dilation And Erosion Of Binary Images Algorithm For Dilation: { For each background pixel (which we will call the input pixel) we superimpose the structuring element from the top of the input image so that the origin of the structuring element coincides with each input pixel position.

14 33 If for every pixel in the structuring element, the corresponding pixel in the image underneath is a foreground pixel, then the input pixel is left as it is. If any of the corresponding pixels in the image are background, the input pixel is also set to foreground value. End after the structuring spans whole of the input image. } Algorithm For Erosion: { For each background pixel (which we will call the input pixel) we superimpose the structuring element from the top of the input image so that the origin of the structuring element coincides with each input pixel position. If for every pixel in the structuring element, the corresponding pixel in the image underneath is a foreground pixel, then the input pixel is left as it is. If any of the corresponding pixels in the image are background, the input pixel is also set to background value. End after the structuring spans whole of the input image. } 3.5 MORPHOLOGICAL PROCESSING OF 2-D GRAY / COLOR IMAGES Binary Morphological operations are easily extendable to grey scale images using minimum and maximum operations. Erosion (dilation) of an image is the operation of assigning to each pixel the minimum (maximum) value found over a neighbourhood of the corresponding pixel.

15 Morphological Erosion of 2-D Gray / Color Images Erosion of an input gray image A(m,n) by the structuring element B defined by Serra, J., Academic Press, London et al (1982) and Craig Howard Richardson in symbolic form as A B m,n = min A m, n + b = G m,n b B min (3.27) g Bx Where m is height and n is width. The gray-scale erosion is a point wise minima of the image over the region of original image of the translated structuring element Algorithm for Erosion of Grayscale and Color Images Repeat sliding the structuring element over the image { subtract pixels of structuring element from the corresponding pixels of the image find the minimum, k, among all of them if all the structuring element pixels are less than the corresponding image pixels then replace the central pixels in the image with k; else replace it with 0 } until the structuring element spans whole of the image Morphological Dilation Of 2-D Gray / Color Images Dilation of an input gray image A(m,n) by the structuring element B is defined by Serra, J., Academic Press, London 1982 and Craig Howard Richardson in symbolic form as

16 35 A B m,n = max A m, n + b = G m,n b B max (3.28) g Bx The gray-scale dilation is a point wise maxima of the image over the region of original image of the translated structuring element. Algorithm for Dilation of Grayscale and Color Images Repeat sliding the structuring element over the image { Add pixels of structuring element to the corresponding pixels of the image find the maximum, k, among all of them if at least one of the image pixels that are spanned by structuring element is non-zero then replace the central pixels in the image with k; else replace it with 0 } until the structuring element spans whole of the image. Opening and Closing Of Images We can combine dilation and erosion to build two important higher order operations Opening and Closing. Opening is erosion followed by dilation and defined as, AoB = (A B) B (3.29) Closing is dilation followed by erosion and defined as, A B = (A B) B (3.30)

17 MORPHOLOGICAL PROCESSING OF 3-D IMAGES Mathematical morphology is a set theoretic and non-linear way of processing the digital images. The main feature of the mathematical morphology which makes it popular is analyzing images based on its geometry. Some of the analytical capabilities of this concept are size or shape descriptions, finding spatial relationships between different objects, and connected component labeling based on the topological properties of the object. Mathematical morphology was originally developed for binary images, viewed as subsets of the integer grid Z 2 (or Z d, for any dimension d), and was later extended to gray scale images and multi-band images and 3-D images. The two most basic operations in mathematical morphology are erosion and dilation. Both of these operators take two pieces of data as input, an image to be eroded or dilated, and a structuring element (also known as a kernel). The two pieces of input data are each treated as representing sets of coordinates in a way that is slightly different for binary and gray scale images. The binary and gray scale morphological operations are discussed by Serra et al (1982) and Gonzalez and Woods et al (2002). The 3-D morphological operations are derived from 2-D mathematical morphological operations by using 3-D structuring elements. This concept was initiated by Rajan et al (1990) and Jirawit Lerdsinmongkol et al (2008). Suppose A is a 3-D image with mxnxd dimension, where m is width, n is height and r is number of layers of an input dataset (For example in our case it is ). B is a structuring element with k k k dimension. For example in our case the structuring element size is i.e. 27 neighbourhood including central pixel as shown in Figure 3.4. There are some other 3-D structuring elements such as 7-neighbourhood, 9-

18 37 neighbourhood, 15-neighbourhood and 19-neighbourhood as shown in the Figure 3.5. Figure neighborhood Structuring Element (a) (b) (c) (d) Figure 3.5 Other types of structuring elements [a] 7-Neighbourhood, [b] 9-Neighbourhood, [c] 15-Neighbourhood, [d] 19-Neighbourhood The 3-D mathematical operations are performed by convolving the 3-D structuring element B on the original image A Erosion is defined by the following equation.

19 38 A B = min (a ) (i, j,k) b x+i, y+ j, c+k (3.31) Similarly, dilation is defined by the following equation. A B max (A ) (i, j,k) B a i, b j, c k (3.32) and erosion. One can realize the operations of closing and opening using dilation Image A is closed by the image B using the following equation. A B (A B) B (3.33) Image A is opened by the image B using the following equation. A B (A B) B (3.34) Morphological Erosion of 3-D Gray Images Erosion of a 3-D gray scale image A(m,n,r), where m is height, n is width and r is depth, by the structuring element B is defined by Serra, J., Academic Press, London 1982 and Craig Howard Richardson in symbolic form as A B m,n, r = min A m,n,r + b = min G m,n, r (3.35) b B g Bx The structuring element is placed on the first 3-D grid of voxels of the given 3-D image. The output is set to the minimum value lying within the structuring element. The 3-D structuring element is then moved across the 3- D grid in that particular depth, until the entire 3-D image in that depth has

20 39 been processed. Then the structuring element is moved along the depth in the same way. The process is repeated until all the depths in 3-D image have been processed. Algorithm For Erosion repeat sliding the 3-D structuring element over the 3-D image { subtract voxels of structuring element from the corresponding voxels of the image find the minimum, k, among all of them if all the structuring element voxels are less than the corresponding image voxels then replace the central voxel in the image with k; else replace it with 0} until the structuring element spans whole of the image. The above Algorithm for 3-D Erosion of Gray scale is described using symbolic notation Morphological Dilation Of 3-D Gray Images Dilation of a 3-D gray scale image A(m,n,r), where m is height, n is width and r is depth, by the structuring element B is defined by Serra, J., Academic Press, London 1982 and Craig Howard Richardson in symbolic form as A B m,n,r = max A m,n,r + b = max G m,n,r (3.36) b B g Bx The structuring element is placed on the first 3-D grid of voxels of the given 3-D image. The output is set to the maximum value lying within the structuring element. The 3-D structuring element is then moved across the

21 40 3-D grid in that particular depth, until the entire 3-D image in that depth has been processed. Then the structuring element is moved along the depth in the same way. The process is repeated until all the depths in 3-D image have been processed. Algorithm For Dilation repeat sliding the 3-D structuring element over the 3-D image { add voxels of structuring element to the corresponding voxel of the image find the maximum, k, among all of them if at least one of the image voxels that are spanned by structuring element is non-zero then replace the central voxel in the image with k; else replace it with 0 } until the structuring element spans whole of the image. Algorithm For Opening The opening operation is obtained by dilating a 3-D image followed by erosion using the same structuring element. Algorithm For Closing The Closing operation is obtained by eroding the images followed by dilation using the same structuring element.

22 41 Hardware used The volume processing of 3-D images is carried out in a PC having the following configuration: CPU: Intel Core 2 Duo E4400 with 200GHz Speed, 2GB RAM 320GB hard disk. Operating system: Microsoft XP Professional with Service Pack 2. Four different GPU Cards are used to implement volume rendering and processing. Out of four, three GPU s make use of the same desktop system configuration mentioned above. Those three GPU s are NVIDIA GeForce 6200 Turbo cache, NVIDIA GeForce 8800 GT and NVIDIA GeForce 9800 GT. The other GPU i.e. ATI Radeon HD 5970 uses a desktop PC fitted with AMD Athlon 64x2 dual core processor running at speed 3.11 GHz with 2GB RAM and Microsoft XP Professional with service pack 2. Programming environment The programs for volume rendering and volume processing have been developed in C# language of Microsoft Visual Studio 2008 supported by DirectX9 graphics frame work.

23 42 Sample image of a bonsai tree and its processed versions Dilation: This is a morphological operation used for dilating (bulging) a 3-D image to a desired shape. Figure 3.6(a) shows the CT scanned image of a bonsai tree and Figure 3.6(b) its dilated version. (a) (b) Figure 3.6 (a) 3-D image of a bonsai tree, (b) Dilated version of the image given in (a) Erosion: This is a morphological operation used for eroding (shrinking) a 3-D image. Figure 3.7(a) shows the CT scanned image of a bonsai tree and Figure 3.7(b) its eroded version.

24 43 Figure 3.7 (a) (b) (a) 3-D image of a bonsai tree, (b) Eroded version of the image given in (a) Opening: This refers to the morphological operation of eroding a 3-D image and dilating subsequently. Figure 3.8(a) shows the CT scanned image of a bonsai tree and Figure 3.8(b) its opened version. Figure 3.8 (a) (b) (a) 3-D image of bonsai tree, (b) Opened version of the image given in (a)

25 44 Closing: This refers to the morphological operation of dilating a 3-D mage and eroding subsequently. Figure 3.9(a) shows the CT scanned image of a bonsai tree and Figure 3.9(b) its closed version. Figure 3.9 (a) (b) (a) 3-D image of a bonsai tree, (b) Closed version of the image given in (a) CONCLUSION ` The algorithms pertaining to processing of 2-D and 3-D binary and grey images using morphological operations with the help of certain 2-D and 3-D structuring elements are explained. The structuring element used is of size 3 3 3, for performing the operations of 3-D dilation, erosion, opening and closing. In the next chapter details of 2-D and 3-D structuring elements necessary for carrying out morphological operations are described in details.